Note that the previous estimate holds true as \(\kappa _m > 0\)  as noted in Proposition 3.3.2 of [1]}. Now using Theorem REF  repeatedly for both terms of \(G_{m ,\ell }\)  we get that:
\( \int _{\tau _1}^{\tau _2} \int _{S_{\tau }} | G_{m , \ell } |^2 \, d\mu _{S_{\tau }} d\tau \le C \sum _{k+j = m+1 , k\le m} \int _{\tau _1}^{\tau _2} \int _{S_{\tau }} J^N [ N^k T^j \psi _{\ge \ell } ] \cdot n_{\tau } \, d\mu _{S_{\tau }} d\tau , \) 
